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A191312 Triangle read by rows: T(n,k) is the number of dispersed Dyck paths (i.e., Motzkin paths with no (1,0) steps at positive heights) of length n having abscissa of the first return to the horizontal axis equal to k (assumed to be 0 if there are no such returns). 1
1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 1, 2, 1, 0, 3, 2, 2, 2, 1, 0, 6, 3, 4, 2, 4, 1, 0, 10, 6, 6, 4, 4, 4, 1, 0, 20, 10, 12, 6, 8, 4, 9, 1, 0, 35, 20, 20, 12, 12, 8, 9, 9, 1, 0, 70, 35, 40, 20, 24, 12, 18, 9, 23, 1, 0, 126, 70, 70, 40, 40, 24, 27, 18, 23, 23, 1, 0, 252, 126, 140, 70, 80, 40, 54, 27, 46, 23, 65 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,13

COMMENTS

Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).

Sum_{k>=0} k*T(n,k) = A093387(n+1).

LINKS

Table of n, a(n) for n=0..90.

FORMULA

T(n,0)=1; T(n,1)=0;

T(n,k) = binomial(n-k, floor((n-k)/2))*Sum_{j=0..floor(k/2)-1} c(j), where 2<=k<=n and c(j) = binomial(2*j,j)/(j+1) are the Catalan numbers.

G.f.: G(t,z) = 1/(1-z)+(1-sqrt(1-4*t^2*z^2))/((1-t*z)*(1-2*z+sqrt(1-4*z^2)).

For k>=1, g.f. of column 2k is b_{k-1}*z^{2k}*g and of column 2k+1 is b_{k-1}*z^{2*k+1}*g, where g = 2/(1-2*z+sqrt(1-4*z^2)) and b(k) = Sum_{j=0..k-1} c(j) with c(j) = binomial(2*j,j)/(j+1) = A000108(j) (the Catalan numbers).

EXAMPLE

T(5,3)=2 because we have HUDHH and HUDUD, where U=(1,1), D=(1,-1), H=(1,0).

Triangle starts:

  1;

  1, 0;

  1, 0, 1;

  1, 0, 1, 1;

  1, 0, 2, 1, 2;

  1, 0, 3, 2, 2, 2;

  1, 0, 6, 3, 4, 2, 4;

MAPLE

c := proc (j) options operator, arrow: binomial(2*j, j)/(j+1) end proc: T := proc (n, k) if n < k then 0 elif k = 0 then 1 elif k = 1 then 0 else binomial(n-k, floor((1/2)*n-(1/2)*k))*(sum(c(j), j = 0 .. floor((1/2)*k)-1)) end if end proc: for n from 0 to 12 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form

G := (1-t*z+t^2*z^2*g*C-t^2*z^3*g*C)/((1-z)*(1-t*z)): g := 2/(1-2*z+sqrt(1-4*z^2)): C := ((1-sqrt(1-4*t^2*z^2))*1/2)/(t^2*z^2): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 12 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 12 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form

CROSSREFS

Cf. A001405, A191313.

Sequence in context: A159817 A079532 A328176 * A240159 A309447 A320312

Adjacent sequences:  A191309 A191310 A191311 * A191313 A191314 A191315

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, May 30 2011

STATUS

approved

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Last modified November 14 02:19 EST 2019. Contains 329108 sequences. (Running on oeis4.)